And that never-give-up perseverance -- combined with Ryan Miller's run of splendid goaltending and rookie Marcus Foligno's timely tip-in goal -- has suddenly paid off by lifting the surging Sabres into playoff contention for the first time in three months.
"I think it shows the character we have in the room," captain Jason Pominville said. "The guys stuck together. And as fun as it's been, I think there's still a lot of work to be done."
Buffalo (37-29-10) vaulted two points ahead of the Capitals and alone into eighth place in the Eastern Conference standings.
With six games left, including a critical showdown at Washington on Tuesday, the Sabres aren't in the playoffs yet. But they can at least rest easy for a few days in knowing how they rallied from a flat first period, in which they were being outshot 8-1 at the 6:51 mark, when Minnesota's Steve Kampfer scored.
The Sabres rebounded by outshooting the Wild 36-17 the rest of the way in winning their fourth in a row and improving to 6-0-2 in their last eight and 18-5-5 in their past 28. It's a remarkable run that has Buffalo in eighth place for the first time since the morning of Dec. 20.
Foligno was credited with the go-ahead goal 1:45 into the third period, when Tyler Myers' point shot deflected in off the forward's shin in front. And Miller stone-walled the Wild, making 24 saves to improve to 13-1-3 in his past 17, a stretch in which he's allowed just 31 goals.
"We've got an opportunity to take a breather," coach Lindy Ruff said, noting his team just finished a stretch of playing five games in eight days. "I just told them, 'Get your minds off hockey and get some rest.'"
The Wild (31-33-10) continue to tumble out of playoff contention in dropping to 3-8-1 in their past 12. They're in the midst of an 11-26-7 slide since owning the NHL's best record on Dec. 10.
What's worse for the Wild is that their dreadful run of injuries continued. Initial tests showed that center Matt Cullen broke a finger on his right hand after being struck by a shot midway through the third period.
"I'm so sick of talking about injuries to be honest with you, it's ridiculous," Wild coach Mike Yeo said. "We've still got to play hockey games. It stinks to lose Matt Cullen. It stinks we have guys out of the lineup, but we've got a job to do. Let's make sure we do it."
The Wild began their two-game road swing traveling with an NHL-minimum 20-player roster after defenseman Jared Spurgeon sustained a concussion in a 3-2 shootout win over Calgary on Thursday.
They've also played long stretches minus captain Mikko Koivu, who played his second game, after missing 15 with an upper body injury. And starting goalie Niklas Backstrom missed his 10th game with a lower body injury.
Wild backup goalie Josh Harding was sharp in stopping 34 shots to keep the score close. Harding had no chance on either goal he allowed, and Hodgson sealed the victory by scoring into an empty net with 1 second remaining.
After Vanek tied it during a scramble in front 72 seconds into the second period, Foligno used his big-body presence in front to score the go-ahead goal on the power play, less than a minute after Darroll Powe was penalized for holding.
The Sabres worked the puck around to Myers at the right point, from where he blasted a shot in off Foligno.
Foligno's goal continued a hot stretch for the son of former Sabres captain Mike Foligno. The 20-year-old forward now has six goals and 10 points in eight games since being called up from AHL Rochester.
Ruff wasn't surprised by his team's flat start, considering the Sabres were playing on back-to-back nights after a 4-1 win at the New York Rangers, while the Wild arrived in Buffalo on Friday.
"I wasn't surprised that they had better legs early on, but I thought we dug in," Ruff said. "We came out at the start of the second and started to make a difference."
Miller's best save came 13 minutes into the second period when teammate Christian Ehrhoff carelessly carried the puck through his own crease, only to have Koivu cut him off in front. Koivu got his stick on the puck and directed it toward the net, only to have Miller punch it aside with his glove.
Wild C Kyle Brodziak had an assist to extend his point streak to four games, in which he has two goals and five assists. ... With Backstrom not on the trip, the Wild recalled G Matt Hackett from AHL Houston. ... The Sabres last regulation loss at home was a 4-1 defeat to New Jersey on Feb. 14.